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le bar w delta bar z operatorname cov w_ i z_ i where z displaystyle z represents trait value and w displaystyle w represents fitness either taken for an individual i displaystyle i or averaged over the entire population if fitness is linear in trait value the fitness for an individual i displaystyle i can be written as w i α c z i b z n displaystyle w_ i alpha c z_ i bz_ n where α displaystyle alpha is the component of an individual s fitness which is independent of trait value c displaystyle c parameterizes the effect of individual i displaystyle i s phenotype on its own fitness written negative by convention to represent a fitness cost z n displaystyle z_ n is the average trait value of individual i displaystyle i s neighbours and b displaystyle b parameterizes the effect of individual i displaystyle i s neighbours on its fitness written positive by convention to represent a fitness benefit substituting into the price equation then gives w δ z cov α c z i b z n z i cov α z i c cov z i z i b cov z n z i displaystyle bar w delta bar z operatorname cov alpha cz_ i bz_ n z_ i operatorname cov alpha z_ i c operatorname cov z_ i z_ i b operatorname cov z_ n z_ i since α displaystyle alpha by definition does not covary with z i displaystyle z_ i this rearranges to δ z cov z i z i w b cov z n z i cov z i z i c displaystyle delta bar z frac operatorname cov z_ i z_ i bar w left b frac operatorname cov z_ n z_ i operatorname cov z_ i z_ i c right since cov z i z i w var z i w displaystyle frac operatorname cov z_ i z_ i bar w frac operatorname var z_ i bar w this term must by definition be greater than 0 this is because variances can never be negative and negative mean fitness is undefined if mean fitness is 0 the population has crashed similarly 0 variance would imply a monomorphic population in both cases a change in mean trait value is impossible it can then be said that that mean trait value will increase δ z 0 displaystyle delta bar z 0 when b cov z n z i cov z i z i c 0 displaystyle b frac operatorname cov z_ n z_ i operatorname cov z_ i z_ i c 0 or r b c displaystyle rb c giving hamilton s rule where relatedness r displaystyle r is a regression coefficient of the form cov z n z i cov z i z i displaystyle frac operatorname cov z_ n z_ i operatorname cov z_ i z_ i or cov z n z i var z i displaystyle frac operatorname cov z_ n z_ i operatorname var z_ i 8 relatedness here can vary between a value of 1 only interacting with individuals of the same trait value and 1 only interacting with individuals of a most different trait value and will be 0 when all individuals in the population interact with equal likelihood fitness in practice however does not tend to be linear in trait value this would imply an increase to an infinitely large trait value being just as valuable to fitness as a similar increase to a very small trait value consequently to apply hamilton s rule to biological systems the conditions under which fitness can be approximated to being linear in trait value must first be found there are two main methods used to approximate fitness as being linear in trait value performing a partial regression with respect to both the focal individual s trait value and its neighbours average trait value 9 or taking a first order taylor series approximation of fitness with respect to trait value 10 2 performing a partial regression requires minimal assumptions but only provides a statistical relationship as opposed to a mechanistic one and cannot be extrapolated beyond the dataset that it was generated from linearizing via a taylor series approximation however provides a powerful mechanistic relationship see also causal model but requires the assumption that evolution proceeds in sufficiently small mutational steps that the difference in trait value between an individual and its neighbours is close to 0 in accordance with fisher s geometric model although in practice this approximation can often still retain predictive power under larger mutational steps as a first order approximation linear in trait value hamilton s rule can only inform about how the mean trait value in a population is expected to change directional selection it contains no information about how the variance in trait value is expected to change disruptive selection as such it cannot be considered sufficient to determine evolutionary stability even when hamilton s rule predicts no change in trait value this is because disruptive selection terms and subsequent conditions for evolutionary branching must instead be obtained from second order approximations quadratic in trait value of fitness 2 gardner et al 2007 suggest that hamilton s rule can be applied to multi locus models but that it should be done at the point of interpreting theory rather than the starting point of enquiry 11 they suggest that one should use standard population genetics game theory or other methodologies to derive a condition for when the social trait of interest is favoured by selection and then use hamilton s rule as an aid for conceptualizing this result 11 it is now becoming increasingly popular to use adaptive dynamics approaches to gain selection conditions which are directly interpretable with respect to hamilton s rule 2 altruism edit see also human inclusive fitness the concept serves to explain how natural selection can perpetuate altruism if there is an altruism gene or complex of genes that influences an organism s behaviour to be helpful and protective of relatives and their offspring this behaviour also increases the proportion of the altruism gene in the population because relatives are likely to share genes with the altruist due to common descent in formal terms if such a complex of genes arises hamilton s rule rbc specifies the selective criteria in terms of cost benefit and relatedness for such a trait to increase in frequency in the population hamilton noted that inclusive fitness theory does not by itself predict that a species will necessarily evolve such altruistic behaviours since an opportunity or context for interaction between individuals is a more primary and necessary requirement in order for any social interaction to occur in the first place as hamilton put it altruistic or selfish acts are only possible when a suitable social object is available in this sense behaviours are conditional from the start 12 in other words while inclusive fitness theory specifies a set of necessary criteria for the evolution of altruistic traits it does not specify a sufficient condition for their evolution in any given species more primary necessary criteria include the existence of gene complexes for altruistic traits in gene pool as mentioned above and especially that a suitable social object is available as hamilton noted the american evolutionary biologist paul w sherman gives a fuller discussion of hamilton s latter point 13 to understand any species pattern of nepotism two questions about individuals behavior must be considered 1 what is reproductively ideal and 2 what is socially possible with his formulation of inclusive fitness hamilton suggested a mathematical way of answering 1 here i suggest that the answer to 2 depends on demography particularly its spatial component dispersal and its temporal component mortality only when ecological circumstances affecting demography consistently make it socially possible will nepotism be elaborated according to what is reproductively ideal for example if dispersing is advantageous and if it usually separates relatives permanently as in many birds on the rare occasions when nestmates or other kin live in proximity they will not preferentially cooperate similarly nepotism will not be elaborated among relatives that have infrequently coexisted in a population s or a species evolutionary history if an animal s life history characteristicsusually preclude the existence of certain relatives that is if kin are usually unavailable the rare coexistence of such kin will not occasion preferential treatment for example if reproductives generally die soon after zygotes are formed as in many temperate zone insects the unusual individual that survives to interact with its offspring is not expected to behave parentally 13 the occurrence of sibling cannibalism in several species underlines the point that inclusive fitness theory should not be understood to simply predict that genetically related individuals will inevitably recognize and engage in positive social behaviours towards genetic relatives 14 15 16 only in species that have the appropriate traits in their gene pool and in which individuals typically interacted with genetic relatives in the natural conditions of their evolutionary history will social behaviour potentially be elaborated and consideration of the evolutionarily typical demographic composition of grouping contexts of that species is thus a first step in understanding how selection pressures upon inclusive fitness have shaped the forms of its social behaviour richard dawkins gives a simplified illustration 17 if families genetic relatives happen to go around in groups this fact provides a useful rule of thumb for kin selection care for any individual you often see 17 evidence from a variety of species 18 19 20 including primates 21 and other social mammals 22 suggests that contextual cues such as familiarity are often significant proximate mechanisms mediating the expression of altruistic behaviour regardless of whether the participants are always in fact genetic relatives or not this is nevertheless evolutionarily stable since selection pressure acts on typical conditions not on the rare occasions where actual genetic relatedness differs from that normally encountered 13 inclusive fitness theory thus does not imply that organisms evolve to direct altruism towards genetic relatives many popular treatments do however promote this interpretation as illustrated in a review 23 m any misunderstandings persist in many cases they result from conflating coefficient of relatedness and proportion of shared genes which is a short step from the intuitively appealing but incorrect interpretation that animals tend to be altruistic toward those with whom they share a lot of genes these misunderstandings don t just crop up occasionally they are repeated in many writings including undergraduate psychology textbooks most of them in the field of social psychology within sections describing evolutionary approaches to altruism park 2007 p860 23 such misunderstandings of inclusive fitness implications for the study of altruism even amongst professional biologists utilizing the theory are widespread prompting prominent theorists to regularly attempt to highlight and clarify the mistakes 17 an example of attempted clarification is west et al 2010 24 in his original papers on inclusive fitness theory hamilton pointed out a sufficiently high relatedness to favour altruistic behaviours could accrue in two ways kin discrimination or limited dispersal there is a huge theoretical literature on the possible role of limited dispersal as well as experimental evolution tests of these models however despite this it is still sometimes claimed that kin selection requires kin discrimination furthermore a large number of authors appear to have implicitly or explicitly assumed that kin discrimination is the only mechanism by which altruistic behaviours can be directed towards relatives t here is a huge industry of papers reinventing limited dispersal as an explanation for cooperation the mistakes in these areas seem to stem from the incorrect assumption that kin selection or indirect fitness benefits require kin discrimination misconception 5 despite the fact that hamilton pointed out the potential role of limited dispersal in his earliest papers on inclusive fitness theory 25 24 green beard effect edit see also kin recognition as well as interactions in reliable contexts of genetic relatedness altruists may also have some way to recognize altruistic behaviour in unrelated individuals and be inclined to support them as dawkins points out in the selfish gene 26 and the extended phenotype 27 this must be distinguished from the green beard effect the green beard effect is the act of a gene or several closely linked genes that produces a phenotype allows recognition of that phenotype in others causes the individual to preferentially treat other individuals with the same gene the green beard effect was originally a thought experiment by hamilton in his publications on inclusive fitness in 1964 28 although it hadn t yet been observed as of today it has been observed in few species its rarity is probably due to its susceptibility to cheating whereby individuals can gain the trait that confers the advantage without the altruistic behaviour this normally would occur via the crossing over of chromosomes which happens frequently often rendering the green beard effect a transient state however wang et al has shown in one of the species where the effect is common fire ants recombination cannot occur due to a large genetic transversion essentially forming a supergene this along with homozygote inviability at the green beard loci allows for the extended maintenance of the green beard effect 29 equally cheaters may not be able to invade the green beard population if the mechanism for preferential treatment and the phenotype are intrinsically linked in budding yeast saccharomyces cerevisiae the dominant allele flo1 is responsible for flocculation self adherence between cells which helps protect them against harmful substances such as ethanol while cheater yeast cells occasionally find their way into the biofilm like substance that is formed from flo1 expressing yeast they cannot invade as the flo1 expressing yeast will not bind to them in return and thus the phenotype is intrinsically linked to the preference 30 parent offspring conflict and optimization edit early writings on inclusive fitness theory including hamilton 1964 used k in place of b c thus hamilton s rule was expressed as k 1 r displaystyle k 1 r is the necessary and sufficient condition for selection for altruism where b is the gain to the beneficiary c is the cost to the actor and r is the number of its own offspring equivalents the actor expects in one of the offspring of the beneficiary r is either called the coefficient of relatedness or coefficient of relationship depending on how it is computed the method of computing has changed over time as has the terminology it is not clear whether or not changes in the terminology followed changes in computation robert trivers 1974 defined parent offspring conflict as any case where 31 1 k 2 displaystyle 1 k 2 i e k is between 1 and 2 the benefit is greater than the cost but is less than twice the cost in this case the parent would wish the offspring to behave as if r is 1 between siblings although it is actually presumed to be 1 2 or closely approximated by 1 2 in other w...